Session 3: Scientific Python

An introduction to scientific programming with

Matplotlib tips

Interactive mode:

Clearing the current figure:

or current set of axes:

>>> pyplot.ion()

>>> pyplot.clf()

>>> pyplot.cla()

•  A Python program relevant to your research •  put course material into practice •  opportunity to become familiar with Python •  requirement to qualify for credits

•  Your program should… •  be written as an importable module (.py file) •  do something meaningful: analyse real data or perform a simulation •  use at least two user functions •  use at least one of the modules introduced in Sessions 3 – 5 •  produce at least one informative plot •  comprise >~ 50 lines of actual code (excluding comments and imports)

•  Submit as source code (.py file) and pdf/png files of the plot(s) •  via email (steven.bamford@nottingham.ac.uk) •  by end of 20 December.

•  So far…

•  Session 1:

•  core Python language and libraries

•  Session 2:

•  fast, multidimensional arrays

•  plotting tools

•  Session 3:

•  libraries of fast, reliable scientific functions

•  Suite of numerical and scientific tools for Python

•  http://scipy.org/

•  http://docs.scipy.org/

•  cluster Clustering algorithms •  constants Physical and mathematical constants •  fftpack Fast Fourier Transform routines •  integrate Integration and ordinary differential equation solvers •  interpolate Interpolation and smoothing splines •  io Input and Output •  linalg Linear algebra •  maxentropy Maximum entropy methods •  ndimage N-dimensional image processing •  odr Orthogonal distance regression •  optimize Optimization and root-finding •  signal Signal processing •  sparse Sparse matrices and associated routines •  spatial Spatial data structures and algorithms •  special Special functions •  stats Statistical distributions and functions •  weave C/C++ integration

# scipy submodules

# must be explicitly

# imported, e.g.,

import scipy.fftpack

# or

from scipy import stats

Some simple examples:

•  Special functions (special)

•  Root finding (optimize)

•  Integration (integrate)

•  Statistics (stats)

•  Image processing (ndimage)

•  Interpolation (interpolate)

•  Optimisation (optimize)

•  Huge number of functions, including… •  Bessel functions •  Gamma functions •  Fresnel integrals •  Hypergeometric functions •  Orthogonal polynomials

e.g., Bessel functions of order 1, 2, 3

>>> from scipy import special

>>> x = numpy.arange(0, 10.001, 0.01)

>>> for alpha in range(3):

... y = special.jv(alpha, x)

... pyplot.plot(x, y,

label=r'$J_%i$’%alpha)

>>> pyplot.hlines(0, 0, 10)

>>> pyplot.legend()

•  Accurate automatic root-finding using MINPACK

>>> from scipy.optimize import fsolve # n-dimensional root finder

>>> from scipy.special import jv

Define a function to solve First argument is variable (or array of variables) of interest

>>> def f(z, a1, a2):

... return jv(a1, z) - jv(a2, z)

...

>>> fsolve(f, 2.5, args=(1, 2))

array([ 2.62987411])

>>> fsolve(f, 6, args=(1, 2))

array([ 6.08635978])

>>> pyplot.fill_between(x, special.jv(1, x), special.jv(2, x),

where=((x > 2.630) & (x < 6.086)), color=“peru”)

•  Accurate automatic integration using QUADPACK •  including uncertainty estimate

>>> from scipy.integrate import quad # one-dimensional integration

Using previous function (first argument is variable of interest)

>>> r = fsolve(f, (2.5, 6), args=(1, 2))

>>> print r

[ 2.62987411 6.08635978]

>>> quad(f, r[0], r[1], args=(1, 2))

(-0.98961158607157, 1.09868956829247e-14)

>>> quad(exp, -integrate.Inf, 0)

(1.0000000000000002, 5.842606742906004e-11)

•  Can specify limits at infinity (-scipy.integrate.Inf, scipy.integrate.Inf)

•  QUADPACK and MINPACK routines provide warning messages

•  Extra details returned if parameter full_output=True

>>> quad(tan, 0, pi/2.0-0.0001)

(9.210340373641296, 2.051912874185855e-09)

>>> quad(tan, 0, pi/2.0)

Warning: Extremely bad integrand behavior occurs at some points of the

integration interval.

(38.58895946215512, 8.443496712555953)

>>> quad(tan, 0, pi/2.0+0.0001)

Warning: The maximum number of subdivisions (50) has been achieved.

If increasing the limit yields no improvement it is advised to analyze

the integrand in order to determine the difficulties. If the position of a

local difficulty can be determined (singularity, discontinuity) one will

probably gain from splitting up the interval and calling the integrator

on the subranges. Perhaps a special-purpose integrator should be used.

(6.896548923283743, 2.1725421039565056)

•  Probability distributions •  including: norm, chi2, t, expon, poisson, binom, boltzmann, … •  methods:

•  rvs – return array of random variates •  pdf – probability density function •  cdf – cumulative density function •  ppf – percent point function •  … and many more

•  Statistical functions •  including:

•  mean, median, skew, kurtosis, … •  normaltest, probplot, … •  pearsonr, spearmanr, wilcoxon, … •  ttest_1samp, ttest_ind, ttest_rel, … •  kstest, ks_2samp, …

>>> lambda = 10 >>> p = stats.poisson(lambda)

# P(n > 20) >>> 1 – p.cdf(20) 0.0015882606618580573

# N: P(n < N) = 0.05, 0.95 >>> p.ppf((0.05, 0.95)) array([ 5., 15.])

# true 95% CI bounds on lambda >>> stats.gamma.ppf((0.025, 0.975),

lambda+0.5, 1) array([ 6.14144889, 18.73943795])

>>> x = stats.norm.rvs(-1, 3, size=30) # specify pdf parameters >>> n = stats.norm(1, 3) # create ‘frozen’ pdf >>> y = n.rvs(20) >>> z = n.rvs(50) >>> p = pyplot.subplot(121) >>> h = pyplot.hist((x, y), normed=True,

histtype='stepfilled', alpha=0.5) >>> p = pyplot.subplot(122) >>> h = pyplot.hist((x, y), histtype='step’,

cumulative=True, normed=True, bins=1000)

>>> stats.ks_2samp(x, y) (0.29999999999999999, 0.18992875018013033) >>> stats.ttest_ind(x, y) (-1.4888787966012809, 0.14306062943339182)

>>> stats.ks_2samp(x, z) (0.31333333333333335, 0.039166429989206733) >>> stats.ttest_ind(x, z) (-2.7969511393118509, 0.0064942129302196124)

>>> stats.kstest(x, stats.norm(1, 3).cdf) (0.3138899035681928, 0.0039905619713858087)

•  Additional routines for filtering in ‘signal’ submodule

>>> x = stats.poisson.rvs(0.01, size=(100,100)).astype(numpy.float)

>>> y = ndimage.gaussian_filter(x, 3, mode='constant')

>>> z = y + stats.norm.rvs(0.0, 0.01 , size=(100,100)) >>> zg = ndimage.gaussian_filter(z, 2, mode='constant')

>>> zm = ndimage.median_filter(z, 5, mode='constant')

>>> zu = ndimage.uniform_filter(z, 5, mode='constant')

>>> for i, img in enumerate([x, y, z, zg, zm, zu]):

... ax = pyplot.subplot(2, 3, i)

... pyplot.imshow(img, interpolation=None)

... ax.yaxis.set_ticks([])

... ax.yaxis.set_ticks([])

...

>>> numpy.mgrid[0:5,0:5] array([[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3], [4, 4, 4, 4, 4]],

[[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]]])

>>> numpy.mgrid[0:5:2,0:5:2] array([[[0, 0, 0], [2, 2, 2], [4, 4, 4]],

[[0, 2, 4], [0, 2, 4], [0, 2, 4]]])

>>> numpy.mgrid[0:2:5j,0:2:5j] array([[[ 0. , 0. , 0. , 0. , 0. ], [ 0.5, 0.5, 0.5, 0.5, 0.5], [ 1. , 1. , 1. , 1. , 1. ], [ 1.5, 1.5, 1.5, 1.5, 1.5], [ 2. , 2. , 2. , 2. , 2. ]],

[[ 0. , 0.5, 1. , 1.5, 2. ], [ 0. , 0.5, 1. , 1.5, 2. ], [ 0. , 0.5, 1. , 1.5, 2. ], [ 0. , 0.5, 1. , 1.5, 2. ], [ 0. , 0.5, 1. , 1.5, 2. ]]])

>>> numpy.ogrid[0:2:5j,0:2:5j] [array([[ 0. ], [ 0.5], [ 1. ], [ 1.5], [ 2. ]]), array([[ 0. , 0.5, 1. , 1.5, 2. ]])]

•  Several methods •  most useful are: interp1d, griddata, splrep/splev, bisplrep/bisplev

Create samples

>>> p, q = numpy.random.random_integers(99, size=(2, 1000))

>>> sy = y[p,q]

>>> sz = z[p,g]

>>> grid_x, grid_y = numpy.mgrid[0:100, 0:100]

Plot

>>> vmin = stats.scoreatpercentile(y.ravel(), 0.01)

>>> vmax = stats.scoreatpercentile(y.ravel(), 99.9)

>>> pyplot.imshow(y, interpolation=None, vmin=vmin, vmax=vmax)

>>> pyplot.plot(p, q, ‘.k’)

>>> pyplot.axis([0, 100, 0, 100])

Exact interpolation

>>> gy = interpolate.griddata((p, q), sy, (grid_x, grid_y), method='cubic')

Spline fit to noisy sample

>>> bz = interpolate.bisplrep(p, q, sz, s=0.1, full_output=True)

>>> gz = interpolate.bisplev(numpy.arange(100), numpy.arange(100), bz[0])

Plot

>>> pyplot.imshow(gy[::-1,:], interpolation=None, vmin=vmin, vmax=vmax)

>>> pyplot.imshow(gz, interpolation=None, vmin=vmin, vmax=vmax)

•  Many methods •  downhill and stochastic minimisation •  least squares fitting

Details at http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html

See the exercises for an example.

•  Python wrappers of GNU Scientific Library functions

•  PyGSL: http://pygsl.sourceforge.net/

•  GSL: http://www.gnu.org/software/gsl/

•  Incomplete documentation for Python functions, but almost all of GSL is wrapped, so refer to GSL documentation.

•  Most functionality implemented in SciPy •  Try SciPy first, if you can’t find what you need try PyGSL •  More comprehensive and sometimes more tested, but less ‘Pythonic’ •  e.g. Monte Carlo integration

•  http://rpy.sourceforge.net/

•  Wraps R – a statistics analysis language •  many advanced stats capabilities but quite specialised

•  http://www.sagemath.org/

•  Python-based mathematics software (see next session) •  replacement for Maple, Mathematica

1)  Plot and use fsolve to find the first root of the zeroth-order Bessel function of the second kind, i.e. x where Y0(x) = 0.

2)  Use quad to find the integral of Y0(x) between x=0 and the first root.

3)  Write a function to accurately and efficiently calculate the integral of Y0(x) up to its nth root (remember to ensure fsolve has found a solution). Check for a few n up to n = 100, the integral should be converging to zero.

4)  Use scipy.stats.norm.rvs to create 100 samples from a Normal distribution for some mean and sigma. Then use pyplot.hist to create a 10-bin histogram of the samples (see the return values). Convert the bin edges to values at the centre of each bin.

5)  Create a function f((m, s), a, x, y) which returns the sum of the squared residuals between the values in y and a Gaussian with mean m, sigma s and amplitude a, evaluated at x.

6)  Use function you created in (5) with scipy.optimize.fmin to fit a Gaussian to the histogram created in (4). Plot and compare with scipy.stats.norm.fit.